A Globally Asymptotically Stable Planar Homogeneous Polynomial Vector Field With No Polynomial Lyapunov Function

Abstract

We disprove the conjecture that every globally asymptotically stable homogeneous polynomial vector field admits a homogeneous polynomial Lyapunov function. The counterexample is a planar homogeneous cubic polynomial vector field with integer coefficients. It admits no positive definite homogeneous polynomial with nonpositive Lie derivative and, more strongly, no real-analytic Lyapunov function even locally. Nevertheless, it has an explicit degree-two homogeneous Lyapunov function that is radially unbounded, continuously differentiable everywhere, and smooth away from the origin. We also provide a machine-checked Lean 4 formalization of the main result.